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%I #55 Sep 08 2022 08:45:07
%S 9,1,8,9,3,8,5,3,3,2,0,4,6,7,2,7,4,1,7,8,0,3,2,9,7,3,6,4,0,5,6,1,7,6,
%T 3,9,8,6,1,3,9,7,4,7,3,6,3,7,7,8,3,4,1,2,8,1,7,1,5,1,5,4,0,4,8,2,7,6,
%U 5,6,9,5,9,2,7,2,6,0,3,9,7,6,9,4,7,4,3,2,9,8,6,3,5,9,5,4,1,9,7,6,2,2,0,0
%N Decimal expansion of -zeta'(0).
%C The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - _Rick L. Shepherd_, Mar 08 2014
%C For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - _Jianing Song_, Apr 20 2021
%H G. C. Greubel, <a href="/A075700/b075700.txt">Table of n, a(n) for n = 0..10000</a>
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/WallisFormula.html">MathWorld: Wallis Formula</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogGammaFunction.html">Log Gamma Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingsApproximation.html">Stirling's Approximation</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gamma_function">Gamma function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Normal_curve">Normal curve</a>
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F Equals log(2*Pi)/2 = A061444/2 = log(A019727).
%F Equals Integral_{x=0..1} log(Gamma(x)) dx. - _Jean-François Alcover_, Apr 29 2013
%F More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} (log(Gamma(x)) dx for any t>=0 (the Raabe formula). - _Stanislav Sykora_, May 14 2015
%F Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - _Amiram Eldar_, Aug 21 2020
%e 0.91893853320467274178032...
%p evalf(log(2*Pi)/2,120); # _Muniru A Asiru_, Oct 08 2018
%t Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* _Jean-François Alcover_, Apr 29 2013 *)
%o (PARI) -zeta'(0) \\ _Charles R Greathouse IV_, Mar 28 2012
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // _G. C. Greubel_, Oct 07 2018
%Y Cf. A019727, A061444, A257549.
%K cons,nonn
%O 0,1
%A _Benoit Cloitre_, Oct 02 2002
%E Normalized representation (leading zero and offset) _R. J. Mathar_, Jan 25 2009
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